Is there an alternative to RNNs that doesn't require knowing input history?
You don't necessarily need to roll out the inputs to an RNN; doing so makes it easier to optimize computation (if the sequence length is the same each batch), but it's not a necessity. Furthermore, RNNs (and, incidentally, the brain) doesn't necessarily remember the input history as is; rather, the history is encoded via the RNN's cell state (or states, in the case of LSTMs and other RNN cell architectures with multiple states). Neural Turing Machines (NMTs) and Differential neural computers expand on that concept by also using a larger "memory" storage (in the form of a matrix).